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LIBRARY 


UNIVERSITY  OF  CALIFORNIA 


GIFT  OF 


Class 


LIFE   INSURANCE 


PREMIUMS  AND  RESERVES. 


OF  THE 

UNIVERSITY 


BY 

SHEPPARD    HOMANS, 

ft 

CONSULTING  ACTUARY. 


[COPYRIGHT  BY  THE  SPECTATOR  COMPANY,  i388.J 


1888. 

"THE    SPECTATOR    COMPANY, 

16  DEY  STREET,    NEW   YORK. 


LIFE   INSURANCE   PREMIUMS  AND   RESERVES, 


BY    SHEPPARD    ROMANS,  CONSULTING  ACTUARY. 


The  basis  of  every  sound  system  of  life  insurance  is  the  MORTALITY  TABLE. 
While  nothing  is  more  uncertain  than  the  duration  of  an  individual  life,  the  rates  of 
mortality,. or,  in  other  words,  the  probabilities  of  living  and  dying  in  any  one  year  at 
each  age  among  a  large  number  of  persons  similiarly  situated  as  regards  family  history, 
climatic  influences,  etc.,  can  be  predicted  with  almost  mathematical  precision.  The 
rates  of  mortality  among  insured  lives  at  the  several  ages  have  been  carefully  ascer- 
tained by  observations  among  a  vast  number  of  persons  insured  in  British  and  Ameri- 
can companies.  These  results  are  embodied  in  three  mortality  tables  of  standard 
authority,  viz : 

The  ACTUARIES,  or  COMBINED  EXPERIENCE  TABLE,  deduced  from  the  mortuary 
statistics  of  seventeen  British  companies,  and  published  in  1837. 

The  NEW  ACTUARIES  OR  HM.  TABLE,  deduced  from  the  later  experience  of 
twenty  British  companies,  and  published  in  1869. 

The  AMERICAN  EXPERIENCE  TABLE,  deduced  chiefly  from  the  mortuary  statistics 
of  the  Mutual  Life  Insurance  Company  of  New  York. 

Of  these  the  last  named  table,  confirmed,  as  it  has  been  in  a  remarkable  degree, 
by  the  experience  of  other  American  companies,  is  by  far  the  best  index  of  the  rates 
of  mortality  which  may  be  expected  to  prevail  among  insured  lives  in  the  United 
States.  This  table  has  been  adopted  by  nearly  all  American  companies  as  a  basis 
for  premiums  and  reserves,  and  by  many  States  as  a  standard  of  valuation  for  contin- 
gent insurance  liabilities. 

These  tables  do  not  differ  materially  from  each  other,  and  either  would  be  a  safe 
basis  for  the  transactions  of  American  life  insurance  companies.  Their  teachings  have 
all  the  force  of  natural  laws,  and  these  teachings  cannot  be  disregarded  or  violated 
with  impunity. 

Columns  (i)  and  (2)  of  the  following  Table  No.  i,  show  respectively  the  numbers 
living  and  dying  at  each  successive  age  out  of  100,000  persons  starting  at  the  age  of 
ten  years.  Column  (3)  shows  for  each  age  the  rate  of  mortality,  or  probability  of 
dying  within  one  year.  This  is  also  the  cost,  without  interest,  to  insure  one  dollar,  or 
unity,  payable  in  case  of  death  within  the  year,  and  is  found,  for  any  age,  by 
dividing  the  number  of  deaths  by  the  number  living.  For  instance,  at  age  40, 
dividing  765,  the  number  dying,  by  78,106,  the  number  living,  we  have  .009794  as  the 

112686 


TABLE  No.   i. 


Probability  of  Dying 
at  Each  Age,  Which 

Probability  of 
Living 

Cost  to  Insure 
in  case  of  Deatl 

$1,000  Payable 
i.    Am,  Exp.  456. 

T   •      * 

.           . 

is  Also  the  Cost  to 

Through  the 

AGE. 

X 

Number  Living  at 
Each  Age. 

JN  umber  Dying 
at  Each  Age. 

Insure  $1.00  lor  One 
Year,  at  Each  Age. 

Year  at  Each 
Age. 

For  One  Year 

Equal  Yearly 
Premiums  Dur- 

dx 

x  dx 

Only,  at  Age 

ing  Remain- 

/x 

dx 

'* 

i-T 

X 

der  of  Life. 

(I) 

(2) 

(3) 

(4 

(5) 

(6) 

10 

100,000 

749 

.007490 

.992510 

7  20 

10.53 

II 

99.250 

746 

.007516 

.992484 

7.23 

10.70 

12 

98,505 

743 

.007543 

•992457 

7.25 

10.88 

13 

97,762 

740 

.007569 

.992431 

7.28 

ii.  06 

14 

97,022 

737 

•007596 

.992404 

7-30 

11.26 

15 

96,285 

735 

.007634 

.992366 

7.34 

11.47 

16 

95.550 

732 

.007661 

•992339 

7-37 

11.69 

17 

94,818 

729 

.007688 

.992312 

7-39 

11.91 

18 

94,089 

727 

.007727 

.992273 

7-43 

12.15 

19 

93,362 

725 

.007765 

•992235 

7-47 

12.40 

20 

92,637 

723 

.007805 

.992195 

7-51 

12.67 

21 

91,914 

722 

.007855 

.992145 

12  95 

22 

91,192 

721 

.007906 

.992094 

7.60 

13.24 

23 

90471 

720 

•007958 

.992042 

7-65 

13-55 

24 

89,751 

719 

.008011 

.991989 

770 

13-87 

89,032 

718 

.008065 

•991935 

7-75 

14.21 

26 

88,314 

718 

.008130 

.991870 

7.82 

14-57 

28 

87.596 
86,878 

718 
718 

.008197 
.008264 

.991803 
.991736 

7.88 
7-95 

14-95 
15-35 

29 

86,160 

719 

.008345 

•991655 

8.02 

15-77 

3° 

85,441 

720 

.008427 

•991573 

8.10 

16.21 

31 

84,721 

721 

.008510 

.991490 

8.18 

16.68 

32 

84,000 

.008607 

•99I393 

8.28 

17.18 

33 

83,277 

726 

.008718 

.991282 

838 

17.70 

34 

82,^51 

729 

.008831 

.991169 

8.49 

18.26 

81,822 

732 

.008946 

.991054 

8.60 

18  84 

36 

81,090 

737 

.009089 

.990911 

874 

19  46 

37 

80,353 

742 

.009234 

.990766 

888 

2O.  12 

38 

79,611 

.009408 

.990592 

905 

20.82 

39 

78,862 

756 

.009586 

.990414 

9.22 

21-57 

40 

78,106 

765 

.009794 

.990206 

9.42 

22.35 

41 
42 

76,567 

785 

.010008 
.010252 

•989748 

9  62 
9.86 

23.19 
24.08 

43 

75,782 

797 

.010517 

•989483 

IO.II 

25-03 

44 

74,985 

812 

.010829 

.989171 

10.41 

26.04 

74,173 

828 

.011163 

.988837 

10.73 

27.12 

46 

73-345 

848 

.011562 

.988438 

II.  12 

28.27 

47 

72,497 

870 

.012000 

.988000 

"54 

29.50 

48 

71.627 

896 

.012509 

.987491 

12  03 

30.81 

49 

70,731 

927 

.013106 

.986894 

12.  60 

32.21 

50 
Si 
52 

69,804 
68.842 
67,841 

962 

001 

,044 

.013781 
.014541 
.015389 

.986219 

•985459 
.984611 

13.25 
13.98 
14  80 

33.70 
36.98 

53 

66,797 

,091 

•016333 

.983667 

1571 

38.79 

54 

65,706 

.143 

.017396 

.982604 

40.73 

64-563 

,199 

.018571 

.981429 

17.86 

42  79 

56 

63364 

.260 

.019885 

.980115 

19.12 

45  oo 

57 

62,104 

.325 

•021335 

•978665 

20.52 

47  35 

58 

60,779 

,394 

.022936 

.977064 

22.00 

49  87 

59 

59,385 

,468 

.02472^ 

.975280 

2377 

52.57 

60 

57.917 

,546 

.026693 

•973307 

2567 

55-45 

61 

56,371 

,628 

.C28880 

.971120 

27.77 

58-54 

62 

54-743 

,713 

.031292 

.968708 

30.09 

61.84 

63 

53.030 

,800 

•033943 

.966057 

31.90 

65  39 

64 

51.230 

,889 

.036873 

.963127 

3545 

69.18 

49.341 

,980 

.040129 

.959871 

3859 

73-25 

66 

47.36i 

2,070 

•043707 

•956293 

42.03 

77.61 

67 

45,291 

2,158 

.047647 

•952353 

45-82 

82.28 

68 

43,133 

2,243 

.052002 

.947998 

50.00 

87-29 

69 

40,890 

2,321 

.056762 

•943238 

54-58 

92.65 

70 

38,569 

2.391 

•061993 

.938007 

5961 

98  39 

36,178 

7,448 

.067665 

•932335 

65.06 

104.54 

TABLE  No.   i — Continued. 


Probability  of  Dying 

Probability  of 

Cost  to  Insure  $i,oco  Payable 

at  Each  Age.  Which 

Living 

in  case  of  Death.    Am.  Exp.  4%. 

AGE. 

X 

Number  Living  at    •   Number  Dying 
Each  Age.           |     at  Each  Age. 

is  Also  the  Ccst  to 
Insure  $1.00  for  One 
Year,  at  Each  Age. 

Through  the 
Year  at  Each 
Age. 

For  One  Year 

Equal  Yearly 
Premiums  Dur- 

dx 

i-^L. 

Only,  at  Age 

ing  Remain- 

4                         dx 

T~ 

l-L 

X 

der  of  Life. 

(I)                                       (2) 

(3) 

(4) 

(5) 

(6) 

72 

33,730               2,487 

073733 

.926267 

70.90 

111.13 

73 

31.243               2,505 

.080178 

.919822 

77.09 

118.21 

74 

28,738                 2,501 

.087028 

.912972 

83.68 

125.85 

75 

26,237               2,476 

.094371 

.905629 

90.74 

134.14 

76                    23,761                      2,431 

.102311 

.897689 

9838 

I43-I9 

77                    21,330                      2,369 
78                    18,961                      2,291 

.111064 
.120827 

^879173 

106.79 
116.18 

I53.I4 
164.12 

79                     16,670                     2,196 

•I31734 

.868266 

126.67 

176.30 

80                    14,474                     2,091 

.144466 

•855534 

138.91 

189.87 

81                     12,383                        ,964 

.158605 

.841395 

152-50 

204.95 

82                    10,419                        ,816 

.174297 

.825703 

!67-59 

221  .  82 

83                      8,603                        .648 

.191561 

•808439 

184.19 

240.90 

84                     6,955                       ,470 

•211359 

.788641 

203  23 

262.89 

85                     5485                       .292 

•235552 

•764448 

226.49 

288.62 

86                     4,193                       ,114 

.265681 

•734319 

255-46 

318.82 

87                     3-079                        933 

.303020 

.696980 

291.37 

354-03 

88 

2,140                        744 

.346692 

.653308 

334-13 

394-52 

89                     1.402                        555 

•395863 

.604137 

380.64 

441.22 

90 

847                        385 

•454545 

•545455 

43706 

497.08 

91                        462                        246 

.532466 

•467534 

5"  99 

566.28 

92 

216                        137 

•634259 

•365741 

609.87 

649.34 

93                           79                           58 
94                           21                           18 

•734177 
•857143 

•265823 
.142857 

705.94 
824.18 

736.31 
840.77 

95 

3                            3 

I.OOOOOO 

o.oooooo 

96i54 

961.54 

rate  of  mortality  or  probability  of  dying  within  one  year,  at  that  age.  Column  (4) 
gives  for  each  age  the  probability  of  surviving  through  one  year.  This  is  also  the 
cost,  without  interest,  to  provide  one  dollar,  or  unity,  at  the  end  of  one  year,  payable 
in  case  of  surviving  to  the  end  of  the  year.  This  is  found  by  dividing  the  number 
living  at  the  next  higher  age,  or  one  year  older,  by  the  number  living  at  the  age  indi- 
cated. Thus  for  age  40,  the  probability  of  surviving  through  one  year  is  found  by  divid- 
ing 77>34i,  the  number  living  at  age  41,  by  78,106,  the  number  living  at  age  40,  and  is 
represented  by  the  fraction  .990206.  This  also  is  the  value,  without  interest,  of  one 
dollar,  or  unity,  payable  in  case  a  person  now  aged  40  is  alive  at  the  end  of  one  year. 
As  it  is  certain  that  every  individual  will  be  either  alive  or  dead  at  the  end  of  the 
year,  the  probabilities  of  dying  and  of  living  in  one  year  at  age  40  may  be  represented 
as  follows : 

Probability  of  dying  in  one  year 009794 

Probability  of  living  through  one  year 990206 


Certainty  of  living  or  dying  in  one  year i.oooooo 


Column  (5)  gives  the  cost,  in  advance,  for  each  age  to  secure  $1000  payable  at 
the  end  of  the  year  in  case  of  death  within  the  year,  assuming  interest  at  four  per  cent 


per  annum.  Thus,  for  age  40,  the  sum  of  $9.42  paid  in  advance  is  the  net  cost  to 
secure  $1000  payable  at  the  end  of  the  year  provided  death  should  occur  within  the 
year.  Similiarly  at  age  50,  the  cost  to  insure  $1000  for  one  year  is  $13.25.  At  age 
60,  $25.67  ;  at  age  70,  $59.61,  etc.  This  cost  of  insurance  for  one  year  is,  of  course, 
independent  of  the  form  of  policy  contract,  or  of  the  age  at  which  the  policy  was 
issued,  and  in  general  increases  each  year  as  a  man  grows  older.  These  yearly  in- 
creasing costs  of  insurance  are  called  natural  premiums. 

Z.  It  may  be  laid  down  as  a  fundamental  principle  that  every  life  insurance  company 
must  collect  each  year,  in  some  way,  either  by  direct  payments,  or  partly  from  an 
accumulated  fund  and  partly  by  direct  payments,  the  cost,  according  to  these  natural 
premiums,  to  cover  the  insurance  for  the  year  of  the  net  amount  at  risk  on  each  and 
every  policy  in  force,  based  upon  the  actual  age  attained,  regardless  of  the  age  at 
entry,  the  form  of  policy  contract,  or  the  scale  of  premium  payments,  y. 

These  natural  premiums,  or  cost  of  insurance  for  each  separate  year,  constitute 
the  basis  of  all  sound  life  insurance.  Theoretically,  the  receipt  each  year  of  the 
natural  premium,  or  yearly  cost  of  insuring  the  net  amount  at  risk,  based  always  upon 
the  actual  age  attained,  will  enable  any  company  to  meet  all  its  insurance  obligations  at 
maturity,  on  each  and  every  policy  in  force.  Practically,  it  is  necessary  to  add,  under 
any  form  of  policy  contract,  a  margin  for  necessary  expenses,  and  a  further  margin 
to  guard  against  adverse  contingencies,  such  as  epidemics,  undue  withdrawal  of 
sound  lives,  etc.  But  it  cannot  be  too  clearly  stated  that  natural  premium  payments, 
properly  loaded,  are  not  only  sufficient,  but  are  all-sufficient  to  meet  all  the  insurance 
obligations  of  any  company,  no  matter  what  may  be  the  forms  of  its  policy  contracts 
or  the  methods  of  its  premium  adjustments.  In  fact,  any  payment  in  excess  of  the 
natural  premium  applied  to  the  net  amount  at  risk  and  to  the  actual  age  attained  is 
outside  of,  and  independent  of,  insurance,  and  should  go  to  expenses,  contingent  fund, 
investment  or  surplus.  The  natural  premium  in  any  year  pays  for  the  entire  insurance 
during  that  year,  under  any  and  every  form  of  policy  contract  in  any  and  every  com- 
pany. 

Column  (6)  gives  for  each  age  the  level  or  uniform  premiums,  to  continue  un- 
changed through  the  remainder  of  life,  as  the  consideration  for  securing  $1000  payable 
at  the  end  of  the  year  when  death  occurs.  For  instance,  at  age  40  the  payment 
of  $22.35  annually  in  advance  is  the  net  premium  at  that  age  to  secure  $1000,  pay- 
able at  the  end  of  the  year  when  death  occurs.  These  level  premiums  are  the  com- 
muted equivalents  of  the  natural,  or  increasing  premiums,  as  shown  in  column  (5). 

We  will  now  examine  the  principles  upon  which  these  level  premiums  are  deter- 
mined. 

^The  first  step  is  to  ascertain  the  net  single  premium  or  amount  to  be  paid  down 
in  one  sum  to  secure  $1000  payable  at  death,  whenever  that  event  shall  happen.  It 
is  manifest  that  this  single  premium  is  the  sum  total  of  the  separate  costs  of  insuring 
one  dollar,  or  unity,  in  each  successive  year,  discounted  at  the  rate  of  interest  assumed 
to  the  present  date  or  age.  As  we  have  seen,  the  net  cost  without  interest  at  age  40 


TABLE  No.  2. 


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40 

.009794 

.961538 

.0094177 

1.  000000  x 

1.  000000 

I.OOOOO  — 

O 

41 

.O099IO 

•924556 

.0091620 

.990206 

.961538 

.95212 

I 

42 

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.888996 

.0089348 

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.924556 

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2 

43 

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3 

44 

.010396 

.821927 

.0085449 

.960042 

.854804 

.82065 

4 

45 

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•790315 

.0083781 

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.821927 

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.759918 

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.790315 

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48 

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.928187 
.917049 

.759918 
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.67008 

I 

49 

.011869 

•675564 

.0080179 

•905577 

.702587 

.63625 

9 

50 

.012317 

.649581 

.0080006 

.893709 

•675564 

.60376 

10 

51 

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.624597 

.0080048 

.881392 

.649581 

•57254 

II 

52 

.013366 

.600574 

.0080275 

.868576 

.624597 

.54201 

12 

53 

.013968 

•577475 

.0080663 

.855212 

•600574 

.51362 

13 

54 

.014634 

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.0081257 

.841241 

•577475 

.48580 

14 

.015351 

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.0081960 

.826606 

•555265 

.45899 

15 

56 

.016132 

•513373 

.0082817 

.811257 

•533908 

.43314 

16 

57 

.016964 

.493628 

.0083740 

•795125 

•513373 

.40820 

17 

58 

.017848 

.474642 

.0084712 

.778160 

.493628 

.38412 

18 

59 

.018795 

•456387 

.0085778 

.760313 

.474642 

.36088 

19 

60 

.019794 

•438834 

.0086861 

.741518 

•456387 

•33842 

20 

61 

.020843 

•421955 

.0087950 

.721724 

•438834 

.31672 

21 

62 

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.405726 

.0088983 

.700881 

•421955 

•29574 

22 

63 

.023046 

.390121 

.0089906 

.678949 

.405726 

•27547 

23 

64 

.024185 

•375"7 

.0090722 

•655904 

.390121 

.25588 

24 

65 
66 

.025350 
.026502 

.360689 
.346817 

.0091435 
.0091915 

•631718 
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•375II7 
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.23697 

.21871 

3 

67 

.027629 

•333477 

.0092131 

.579866 

.346817 

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27 

69 

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.320651 
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29 

70 

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.296460 

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•493803 

.308319 

•15225 

30 

.031342 

•285058 

•0089343 

.463191 

.296460 

-I3732 

31 

72 

.031841 

.274094 

.0087275 

.431849 

.285058 

.12310 

32 

73 

.032072 

.263552 

.0084526 

.400008 

.274094 

.10964 

33 

74 

.032021 

.253415 

.0081145 

.367936 

.263552 

.09697 

34 

75 

.O3I7OI 

.243669 

.0077244 

•335915 

•253415 

.08513 

35 

76 

.O3II24 

•234297 

.0072923 

•303515 

.243669 

.07413 

36 

77 

.030331 

•225285 

.0068330 

.273090 

.234297 

.06398 

37 

78 

.029332 

.216621 

.0063539 

.242760 

.225285 

.05469 

38 

79 

.028116 

.208289 

.0058562 

.213428 

.216621 

.04623 

39 

80 

.026771 

.200278 

.0053617 

.185312 

.208289 

.03860 

40 

81 

•025X45   | 

.192575 

.0048424 

•158541 

.200278 

.03175 

82 

.023250 

.185168 

.0043052 

•133396 

.172575 

.02569 

42 

83 

.021100 

.178046 

.0037567 

.110145 

.185168 

.02040 

43 

84 

.018821 

.171198 

.0032221 

.089046 

.178046 

.01585 

44 

85 

.016542 

.164614 

.0027230 

.070225 

.171198 

.OI2O2 

45 

8b  - 

.014263 

.158283 

.0022575 

.053684 

.164614 

.00884 

46 

87 

.011946   i 

.152295 

.0018180 

.039421 

.158283 

.00624 

88 

.009526   i 

.146341 

.0013940 

.027476 

•152295 

.00418 

48 

89 

.007106   ; 

.140713 

.0009999 

.017950 

.146341 

.00263 

49 

90 

.004929 

•I3530I 

.0006669 

.010844 

.140713 

•00153 

50 

QI 

.003150 

.130097 

.0004097 

.005915 

•I3530I 

.OOOSO 

51 

92 

.001754 

.125093 

.0002194 

.002765 

.130097 

.00036 

52 

93 

.000743 

.120282 

.0000893 

.001011 

.125093 

.00013 

53 

94 

.000230 

.115656 

.0000267 

.000269 

.120282 

.00003 

54 

95 

.000038 

.112207 

.0000043 

.000038 

.115656 

II22O7 

.00000 

55 

.  j.  ±  w^sy 

Totals 

"367^747 

1644311 

•  J<J/O/4/ 

i 

' 

to  secure  $i,  payable  at  the  end  of  one  year  in  case  of  death  during  the  first  year, 
is  .009794.  To  find  its  net  present  value,  paid  down,  we  must  discount  this  cost  for 
one  year  at  the  rate  of  interest  assumed.  The  present  value  of  one  dollar,  payable 
certain  at  the  end  of  one  year,  at  four  per  cent  interest,  is  .961538.  The  net  present 
value  of  one  dollar,  or  unity,  payable  at  the  end  of  one  year  in  case  of  death,  on  the 
basis  of  the  American  Table — four  per  cent  interest — is  for  age  40  years  .009794  X 
.961538  =.0094177.  [See  columns  (i),  (2),  and  (3),  Table  No.  2.]  In  the  same  way 
the  net  present  value  of  one  dollar,  or  unity,  payable  at  the  end  of  two  years,  provided 
a  person  now  aged  40  should  die  in  the  second  year,  or  between  ages  41  and  42,  is 
found  by  dividing  774,  the  number  dying,  by  78,106,  the  number  living  at  age  40, 
and  discounting  the  quotient  for  two  years.  Thus  ^loe  =.009910;  this  multiplied  by 
924  556  =.0091620,  and  this  is  the  cost  at  age  40  to  secure  one  dollar,  or  unity,  payable 
at  the  end  of  two  years  in  case  of  death  during  the  second  year.  Again,  the  net  pres- 
ent value  of  one  dollar,  payable  in  case  a  man  now  aged  40  years  should  he  die  in  the 
eleventh  year,  or  between  ages  50  and  51,  is  .0080006.  These  separate  values  are 
shown  in  column  No.  3  in  Table  No.  2.  Their  sum  total  is  .3675747,  and  this  is  the 
net  single  premium  paid  down  to  secure  one  dollar,  or  unity,  payable  at  the  end  of  the 
year,  when  a  person  now  aged  40  years  dies,  whenever  that  event  shall  happen. 

By  a  similar  course  of  reasoning  the  net  present  value  of  one  dollar,  or  unity, 
payable  annually  in  advance  during  the  remainder  of  life  at  any  age,  is  the  sum  total 
of  the  present  values  of  the  separate  chances  of  surviving  during  each  successive  year, 
discounted  to  the  present  date  or  age.  Thus  for  age  40  the  present  value  of  one  dollar 
in  advance  is  unity  or  one  dollar.  The  present  value,  without  interest,  of  one  dollar, 
payable  in  one  year,  or  at  age  41,  is,  as  we  have  seen,  .990206.  This  multiplied  by 
.961538,  the  discount,  gives  .95212  as  the  present  value  of  one  dollar,  payable  at  the 
end  of  one  year,  or  at  age  41,  provided  a  person  now  aged  40  be  then  alive.  The 
present  value  of  one  dollar,  payable  in  ten  years,  or  at  age  50,  provided  a  person  now 
aged  40  be  then  alive,  is  $J?{jJ  =.893 7 09  multiplied  by  .675564  =  .60376.  These  suc- 
cessive net  present  values  are  found  in  column  (6).  Their  sum  total  is  16.44311,  and 
this  is  the  present  value  of  one  dollar  per  annum  in  advance  during  the  lifetime  of  a 
person  now  aged  40  years  upon  the  basis  adopted. 

As  already  shown,  the  net  single  premium  at  age  40  to  secure  one  dollar,  or 
unity,  payable  at  the  end  of  the  year  when  death  occurs,  is  .3675747.  Proportionally, 
a  net  single  premium  of  $16.43311  would  secure  $44.7341  payable  at  death.  But 
$16.44311  is  also  the  net  present  value  at  age  40  of  an  annual  premium  of  one  dollar. 
Therefore,  a  net  level  or  uniform  premium  of  $22.3543  would,  at  age  40,  secure 
$1000  payable  at  death.  [See  column  (6),  Table  No.  i.] 

Let  us  now  suppose  a  company  to  consist  of  78,106  persons,  each  aged  40  years, 
each  insured  for  $1000,  or  $78,106,000  in  all,  and  each  paying  the  net  annual  pre- 
mium of  $22.3543.  The  following  table  No.  3  has  been  prepared  to  show  the  progress 
of  the  fund  each  year  until  the  last  death  claim  has  been  paid  at  the  age  of  96  years, 
on  the  basis  of  the  American  Experience  Table  and  four  per  cent  interest.  Column 

6 


TABLE  No.  3. 
78,106  PERSONS.  AGED  40  YEARS,  INSURED  FOR  $1,000  EACH. 


Share  of 

Each  Per- 

AGE. 

X 

Premiums. 

Fund  at 
Beginning  of 
Year. 

Interest  4%. 

Death  Claims. 

Fund  at  End  of 
Year. 

son  in  the 
Fund  at 
End  of 

Year  or  Net 

Reserve. 

(l) 

(2) 

<3) 

.  (4) 

(5) 

(6) 

4° 

$l,  746,030 

$1,746,030 

$69,840 

$765,000 

$1,050,870 

13-59 

41 

7,728,930 

2,779,800 

111,190 

774.ooo 

2,116,990 

27.65 

42 

1.711,630 

3,828,620 

153-140 

785,000 

3,196,760 

42  18 

43 

1,694,080 

4,890,840 

195,630 

797.000 

4,289,470 

57-20 

41 

1,676,260 

5,965.732 

238,630 

812,000 

5,392,360 

7270 

4^ 

1,658,110 

7,050,470 

282,020 

828,000 

6,504,490 

88.68 

46 

1,639,600 

8,144,090 

325,760 

848,000 

7,621,850 

105  13 

47 

I  620,640 

9,242,490 

369,700 

870,000 

8,742,190 

122.05 

48 

1,601,190 

10,343,380 

413,740 

896,000 

9.861.120 

139.42 

49 

1,581,170 

11,442,290 

457.690 

927,000 

10  972,980 

157.19 

5° 

I  560,440 

12,533,420 

501,340 

962,000 

12,072,760 

175-37 

5i 

1,538,940 

13,611,700 

544.470 

i  001,000 

I3.i55.i70 

I93-9I 

52 

1,516,560 

14,671,730 

586,870 

i  044  ooo 

.  14,214,600 

212.80 

53 

1,493,220 

15,707,820 

628,310 

1,091,000 

15,245,130 

232.02 

54 

1,468,830 

16,713,960 

668560 

1,143.000 

16,239,520 

25I-53 

55 

I  443,290 

17,682,810 

707,310 

1,199000 

17,191,120 

271.30 

5^ 

.   1,416,480 

18,607,600 

744,300 

I  260,000 

18,091,900 

291.31 

57 

1,388  310 

19,480,210 

779  210 

1,325,000 

18.934,420 

3II-52 

58 

1,358,680 

20,293,100 

811  720 

1,394,000 

19.710,820 

33I-9I 

59 

1,327,520 

21,038,340 

841  530 

1,408,000 

20  411,870 

352-43 

60 

1,204,710 

21,706,580 

868,260 

i  546,000 

21,028  840 

373-04 

61 

1,260,150 

22,288,990 

891,560 

1,628,000 

21,552,550 

393-70 

62 

1,223,750 

22,776,300 

911.050 

1.713,000 

21.974,350 

4J4-37 

63 

1,185,450 

23,159,800 

926,390 

1,800,000 

22,286,190 

435-01 

64 

I  145,210 

23,431,400 

937,260 

1,889  0°° 

22,479,660 

455-59 

6^ 

1,102,480 

23,582,140 

943,280 

1,980,000 

22,545,420 

476.03 

66 

1,058.720 

23,604.140 

944.i6o 

2,070.000 

22,478,300 

496.31 

67 

1,012,450 

23-490-750 

939.630 

2,158,000 

22,272,380 

516.36 

68 

964,210 

23,236,590 

929,460 

2,243,000 

21,923.050 

536i5 

69 

914,070 

22,837,120 

913,490 

2  321,000 

21,429,610 

555-62 

70 

862,180 

22,291,790 

891  670 

2,391,000 

20,792,460 

574-73 

7i 

808,740 

2I,6oi,22O 

864,050 

2,448,000 

20,017,270 

593-45 

72 

754-01° 

20,771,280 

830,850 

2,487,000 

19.115,130 

611  82 

73 

698,420 

I9.8i3,550 

792,540 

2,505,000 

18,101,090 

629.86 

74 

642,420 

18,743,510 

749-740 

2,501,000 

16,992.250 

64764 

75 

586,510 

17,578,760 

703,150 

2,476.000 

15,805,910 

665.^0 

76 

53i,i70 

16,337,100 

653-480 

2,431,000 

14.559-580 

68258 

77 

476,830 

15,036,410 

601,460 

2,369,000 

I3,2b8,870 

699.79 

78 

423,870 

13,692,740 

547.710 

2,291,000 

11,949,450 

716.82 

79 

372,650 

12,322,110 

492,880 

2,196,000 

10,618,980 

733-65 

80 

32356o 

10,942,540 

437-700 

2,091,000 

9,289,240 

75097 

81 

276,820 

9,566.060 

382,640 

i  964  ooo 

7,984,700 

766.36 

82 

232  910 

8,217,610 

328,700 

I  816,000 

6  730,310 

78232 

83 

192,320 

6,922,630 

276,900 

1,648,000 

5  55L530 

798.20 

84 

I55,48o 

5,707,010 

228.280 

1.470,000 

4,465,290 

814  10 

8=5 

122,620 

4,587,910 

183520 

1,292,000 

3,479.430 

829  82 

86 

93.74° 

3.573.170 

142,930 

1.114,000 

2,602,100 

84479 

87 

68,630 

2,670,730 

106,830 

933-000 

1,844  560 

859-54 

88 

47-980 

1,892,540 

75-900 

744,000 

1,224  240 

87321 

89 

3L340 

1,255,580 

50,220 

555.ooo 

750,800 

886.42 

90 

18,940 

769,740 

30,790 

385,000 

415.530 

899.42 

91 

10,330 

425,860 

17830 

246,000 

196,890 

9"-53 

92 

4,830 

201,720 

8.070 

137000 

72,790 

921.39 

93 

1,770 

74.56o 

2.980 

58,000 

19.540 

93049 

9^ 

470 

26,010 

800 

18,000 

2,810 

936.67 

QC 

7° 

2,880 

1  20 

o  OOO 

IOOO.OO 

yj 

J,V-*JV 

(i)  shows  the  total  premiums  paid  by  those  alive  at  the  beginning  of  each  successive 
year.  Column  (2)  shows  the  fund  at  the  beginning  of  each  year  just  after  the  pre- 
miums have  been  paid.  Column  (3)  shows  the  interest  on  the  fund  each  year.  Column 
(4)  shows  the  death  claims  in  each  year.  Column  (5)  shows  the  fund  at  the  end  of  each 
successive  year.  Column  (6)  shows  the  share  held  for  account  of  each  survivor  in  each 
successive  year  (found  by  dividing  the  total  fund  by  the  number  of  persons  surviving), 
and  this  is  also  the  net  investment  reserve  upon  each  policy. 

The  functions  of  the  investment  reserve  will  be  made  clearly  apparent  by  a  study 
of  Table  No.  4,  which  has  been  prepared  to  illustrate  the  appropriation  each  year  of  the 
component  parts  of  an  ordinary  whole  life  level  premium  of  $313,  paid  annually  in  ad- 
vance, to  secure  $10,000  at  the  death  of  a  man  now  aged  40  years  (or,  rather,  at  the 
end  of  the  year  when  death  occurs).  Column  (i)  shows  the  net  reserve  at  the  end  of 
each  successive  year.  Column  (2)  shows  the  corresponding  net  amount  at  risk  borne 
by  the  company  during  each  successive  year.  This  is  always  the  difference  between 
the  face  of  the  policy  and  the  net  reserve,  which  last,  being  in  hand,  is  not  subject  to 
any  insurance  risks.  Column  (3)  shows  the  net  cost  to  insure  $10,000  during  each 
separate  year  by  the  scale  of  natural  premiums,  as  indicated  in  column  (5),  Table  i. 
Column  (4)  shows  the  cost  to  insure  the  net  amounts  at  risk  at  the  successive  ages  in- 
dicated in  the  margins.  Column  (5)  shows  the  deposit  portion  of  the  annual  premium 
in  each  year,  which,  until  the  age  of  68  is  attained  in  the  example  given,  goes  to  swell 
the  investment  reserve  or  accumulated  deposit.  After  the  age  of  68  the  yearly  costs  to 
insure  the  net  amount  at  risk  exceed  the  entire  net  premiums,  and  hence  the  deficien- 
cies (as  indicated  by  the  minus  sign)  must  be  supplied  by  drawing  from  the  reserve 
fund. 

From  the  foregoing  it  will  be  apparent : 

(i.)  Every  level  premium  policy  is  in  reality  a  contract  for  a  yearly  decreasing 
amount  of  insurance,  and  a  yearly  increasing  amount  of  investment.  It  is  a  combina- 
tion of  insurance,  which  is  one  thing,  with  investment,  which  is  quite  another  thing. 
There  is  no  necessary  connection  between  the  two.  Insurance  or  indemnity  may  be 
purchased  without  investment,  as  investment  may  be  purchased  without  insurance. 
The  investment  element  does  not  add  to  the  security  of  the  insurance,  the  yearly  cost 
of  which  depends,  under  any  and  every  form  of  policy,  upon  the  net  amount  at  risk 
borne  by  the  company,  and  the  actual,  present,  attained  age  of  the  person  whose  life 
is  exposed  to  mortality.  For  instance,  in  the  example  given  (Table  No.  4)  of  a  whole 
life  insurance  policy  of  $10,000,  issued  at  the  age  of  40,  the  reserve  or  invested  de- 
posits, at  the  end  of  twenty  years,  or  at  age  60,  is  $3,730.35.  Now,  this  sum  is  in 
hand,  and  is  not  subject  to  any  insurance  hazard,  hence  the  net  amount  at  risk  for  that 
year  is  $6,269.65  only.  The  cost  to  insure  $10,000  for  one  year  at  age  60,  as  shown 
in  column  (3),  is  $256.67.  Proportionately  the  cost  to  insure  $6,269.65,  the  net 
amount  at  risk,  is  $160.92,  and  this  is  all  the  insurance  done  by  the  company  with  re- 
spect to  that  policy  during  that  year.  At  age  70  the  net  amount  at  risk  is  only 
$4,254.74,  the  cost  of  which  for  that  year,  $253.50,  is  $29.96  more  than  the  net  an- 

8 


TABLE  No.  4. 

WHOLE   LIFE    INSURANCE   BY   LEVEL  OR  UNIFORM    PREMIUMS.     AGE   AT   ISSUE   40    YEARS 
AMOUNT  INSURED  $10,000.     ANNUAL  PREMIUM  DURING  LIFE,  $313. 


AGE. 

Net  Reserve  or  Accu- 
mulated De  p  o  s  i  ts, 
being  ^//"-Insurance 
at  End  of  Year. 

Net  Amount  of  Insur- 
ance Carried  by  the 
Company  During  the 
Year. 

Tabular  Cost  to  Insure 
$10,000  During  Each 
Year.  Am.  Exp. 
Table  4  per  cent. 

Ditt  >,  to  Insure  the  Net 
Amount  at  risk  Each 
Year,  being  also  the 
Full  Insurance  Re- 
serve each  Y»-ar. 

Deposit  Portion  of  j 
each  Premium  which 
is  merely  for  Accu  • 
mulation. 

o  v 
,2 

|l 

«|§ 

llf 

•3  l-ti 

4O 

$I3S  83 

(2) 

$9  864  12 

$q4  1  8 

,  (4) 

$Q2  QO 

(6) 
S8o  46 

(7) 

41 

276  40 

0  723  CT 

q6  23 

nq  cq 

80  46 

313  OO 

42 

421  83 

q  578  17 

q8.C.8 

q4  42 

8q  46 

313  OO 

43 

572  04 

q,427  q6 

101.13 

oc,  qi 

128  20 

8q  46 

qiq  oo 

44 

726.98 

q,273  02 

104.12 

q6  SS 

126  qq 

8q  46 

qiq  OO 

4S              .... 

88682 

9,113  08 

107  34 

q7  82 

I2C  72 

80  46 

qiq  oo 

46 

1,051.31 

8,948  69 

III.I7 

00  48 

124  06 

80  46 

qiq  OO 

47  

1,220.50 

8,779.50 

115  39 

101.31 

122  23 

80  46 

313  oo 

48.  . 

1,394.15 

8,605.85 

120.28 

103.51 

1  20  03 

8q  46 

313  oo 

40  

1,571.94 

8,428.06 

126.02 

106  21 

117  33 

8q  46 

313  oo 

I  7Sq  66 

8,246  34 

132  51 

ICQ  27 

114  27 

8Q  46 

313  OO 

CJ 

I  q3Q  o3 

8,060  92 

I3q  8l 

H2  70 

1  IO  84 

8O  46 

313  OO 

2,127  OQ 

7  872  01 

147.07 

116  48 

107  06 

80  46 

qiq  oo 

CO 

2,320  16 

157  05 

120  61 

IO2  q3 

80  46 

qiq  oo 

CA           

2,51s  2S 

7,484  75 

167  27 

125  20 

O8  34 

80  46 

qiq  oo 

ecr          

2,713  O2 

7,286.98 

178  57 

130  12 

93  42 

80  46 

qiq  oo 

si.  '.'.. 

2,913.10 

7,086.90 

191.20 

I35-5O 

89  46 

qtq  OO 

cy  .    .       ... 

3  115.22 

6  884.78 

205.15 

141.24 

8  •  30 

8946 

3I3.OO 

58 

3  3IQ  OQ 

6  680  91 

220  03 

147  OO 

80  46 

qiq  oo 

Co  

3,524.25 

6,475.75 

237-69 

IS3-93 

6q6i 

89.46 

3I3.OO 

60... 

3,730.35 

6,269.65 

256.67 

'  160  92 

6262 

89.46 

313  oo 

61 

3,036  os 

6,063.05 

277.69 

160  3S 

S4  iq 

8q  46 

qiq  oo 

62 

4,143  66 

5,856.34 

300  88 

176  20 

47  34 

8q  46 

qiq  oo 

63 

4,350.12 

5,649  88 

318.95 

1  80  20 

43  34 

8q  46 

qiq  oo 

64" 

A  555  86 

354.54 

103.  OI 

3O  S3 

8q  46 

qiq  oo 

4,760  33 

5,239.67 

385.85 

202  1  8 

21  36 

89  46 

qiq  CO 

66 

4,963  07 

5,036.93 

420,26 

2ii  68 

11.86 

8q46 

qiq  oo 

67 

5,163  64 

4.836.36 

458.15 

221.^8 

I.  q6 

8q  46 

313  oo 

68 

5,361.46 

4,638.54 

500.02 

231.04 

—  8.4O 

89.46 

313  oo 

60... 

5,556.16 

4,443.84 

545.79 

242  53 

—  iS.QO 

80.46 

313  oo 

7O 

5,747  26 

4,2^2.74 

596.08 

2S3  SO 

—  2Q  06 

80  46 

313  OO 

71 

4,065  46 

650  63 

264  61 

—  41  O7 

8q  46 

qiq  oo 

72  ....  ..... 

6,118  19 

3.881.81 

708  97 

27S  23 

—  CT  60 

8q  46 

qiq  OO 

70 

6,298.64 

3,701.36 

770  94 

285.35 

—  61  81 

8q  46 

qiq  oo 

74.  .         .... 

6,476.42 

3,523  58 

836.80 

2Q4  85 

—  71  31 

8q  46 

313  oo 

7cr  

6,652.02 

3,347.98 

907.41 

3O3.8O 

—80  26 

8q  46 

313  oo 

76  :.::; 

6.825.83 

3,174  17 

983  76 

312.26 

—88  72 

89.46 

3I3.OO 

77 

6  QO7  Q3 

3,002.07 

1,067  03 

32O  60 

—  Q7  06 

8q  46 

313  OO 

78  

7,168.17 

2,831.83 

1,161.80 

329.OO 

y/>v~ 

—  105.46 

89.46 

313.00 

70 

7,336.51 

2,663.49 

1,266.67 

337  22 

—  113  68 

80  46 

qiq  oo 

80 

7,509.70 

2,490  30 

1,389  10 

34  C,  qo 

—  122  36 

8q  46 

qiq  OO 

7,663  60 

2.336.40 

1,525.04 

356  31 

—  132  77 

8q  46 

313  oo 

82  

7,823.20 

2,176.80 

1,675.93 

364  83 

—  14!  29 

89  46 

313  oo 

7,982  oo 

2,018.00 

1,841.93 

371  70 

—  140  l6 

89.46 

3I3.OO 

84  . 

8,141  oo 

1,859  °° 

2,032.30 

377  8  1 

—  154.27 

89.46 

313.00 

85  

8  298,20 

1,701.80 

2,264  92 

385.44 

—  161  90 

89.46 

313.00 

86  

8,447.90 

1,552.40 

2,554,62 

396.57 

—  173.03 

89.46 

313,00 

8,595.40 

1,404.60 

2,913.66 

4Oq  26 

—  i8s  72 

80  46 

qiq  oo 

88 

8,732.10 

1,267.90 

3,335.57 

422  61 

—  IQQ  O7 

8q  46 

3I3.OO 

80... 

8,864.20 

1,135.80 

3,806.38 

432  32 

—  208  78 

8q  46 

313.00 

QO  .  . 

8,994.20 

1,005  80 

4,370  63 

430  60 

—  216.06 

8946 

313.00 

91  

9,115.30 

884.70 

5,11988 

452.96 

—  229.42 

89.46 

313.00 

Q2 

0,213.00 

786.10 

6,098.68 

—  2SS  88 

8q  46 

qiq  oo 

03 

0,304.00 

695,10 

7,059.40 

4QO  6q 

267  15 

8q  46 

qiq  oo 

04  .    .... 

9,366,70 

633.30 

8,241.76 

208.42 

89  46 

313  oo 

QC  

10,000.00 

9,615.40 

3 

8q.46 

313.00 

nual  premium  ($223.54).  The  deficiency  for  that  year,  as  well  as  the  deficiencies  for 
each  subsequent  year,  as  shown  in  column  (5),  must  be  met  by  drawing  on  the  in- 
vestment reserve,  or  accumulated  fund,  the  express  functions  of  which  is  to  provide  for 
the  excessive  cost  of  insurance  in  old  age  when  the  level  premium  is  insufficient  for 
that  purpose. 

(2).  The  investment  reserve  is  occasioned  solely  by  the  artificial  condition  in  the 
level  premium  contract,  which  provides  that  the  premiums  shall  not  increase  as  the 
insured  grows  older,  and  to  enable  the  company  to  pay  the  sum  insured  as  an  endow- 
ment. 

(3).  Whether  the  combination  of  insurance  and  investment  is  desirable  or  advan- 
tageous, depends  upon  the  manner  in  which  each  is  administered.  If  either  the  in- 
surance or  the  investment  can  be  obtained  on  better  terms  separately,  the  combina- 
tion of  the  two  is  certainly  undesirable  and  disadvantageous  to  the  policyholder. 

Instead  of  contracting  with  a  life  insurance  company  for  both  insurance  and  in- 
vestment, which  together  make  up  the  sum  insured,  two  separate  contracts  might  be 
made — the  one  with  a  life  company  for  the  yearly  decreasing  amounts  of  insurance 
only,  see  column  (2)  table  4,  the  other  with  a  savings  bank  or  trust  company  for 
accumulating  the  deposit,  or  investment  portions  of  the  yearly  premium,  see  column  (5) 
of  the  same  table.  In  case  of  death  in  such  case  the  insurance  company  would  pay 
the  net  amount  insured  only,  column  (2),  while  the  savings  bank  would  pay  the  ac- 
cumulated deposits,  column  (i),  the  two  together  making  up  the  full  amount  guar- 
anteed. 

To  show  even  more  clearly  how  the  insurance  and  investment  elements  may  be 
completely  separated  the  following  tables  have  been  prepared. 

Table  No.  5  illustrates  the  case  of  an  endowment  assurance  issued  at  age  of  forty 
years  for  $10,000  payable  in  ten  years  or  at  death  if  prior.  The  net  premium  only 
($853.62)  is  considered — the  margin  for  expenses  and  adverse  contingencies  being  dis- 
regarded. 

Tables  6  and  7  are  intended  to  show  how  the  same  result  can  be  secured  by 
purchasing  a  ten-year  term  insurance  with  the  insurance  company,  annual  premium 
$106.03,  and  a  pure  endowment  (payable  only  in  case  of  survival)  by  depositing  the 
residue  ($747.59)  of  the  endowment  assurance  premium  for  accumulation.  In  case  of 
death  at  any  time  during  the  ten  years,  the  insurance  company  would  pay  the  full 
amount  insured,  and  the  endowment  fund  would  be  lost.  In  case  of  surviving,  the 
$10,000  would  be  paid  as  an  endowment,  and  the  insurance  would  cease. 

The  same  principles  apply  to  any  other  term  of  years,  as  a  whole  life  policy  is  in 
reality  an  endowment  assurance  payable  on  attaining  the  age  of  ninety-six  years,  or 
at  death  if  prior. 


10 


Comparison  of  an  endowment  assurance  contract,  a  ten  year  term  level  premium 
contract,  and  a  pure  endowment  contract.     Amount  $10,000,  and  age  at  issue  40 

years,  in  each  case : 

TABLE  No,  5. 
ENDOWMENT  ASSURANCE,  ANNUAL  PREMIUM  $853.62. 


YEAR. 

Net  Reserve 
or  Accumulated 
Deposits  Being 
Self-Insurance. 

Net  Amount  of 
Insurance  at 
Risk  or  Carried 
by  the  Company. 

Tabular  Cost 
Each  Year 
to  Insure 
$10,000  for 
the  Year. 

Tabular  Cost  to 
Insure  Net 
Amount  at  Risk 
which  is  also 
the  Full  Legal 
and  Mathemat- 
ical Insurance 
Reserve. 

Deposit 
Portion  of 
Annual 
Premium 
Which  is 
Merely   for 
Accumu- 
lation. 

I                          , 

$797.63 

$9,202  37 

$94  1  8 

$86.67 

$766  95 

2                                  

1,633  57 

8,36643 

96  23 

80.51 

773-H 

3..  

2,509  89 

7,490.11 

98.58 

73-84 

779.78 

o  428.0^ 

6,571.05 

101.13 

6645 

787  17 

4  .  QQQ.I6 

5,606.84 

104.12 

58.38 

795-24 

§ 

<t  4<X.q6 

4,494  64 

107.34 

48.24 

805.38 

6,468.51 

3,531.49 

111.17 

39.26 

814.36 

g 

7,586.05 

2,413.95 

115.39 

27.85 

825.77 

8,761.76 

1,238.24 

120.28 

14.89 

838.73 

10,000.00 

Nil. 

126.02 

Nil! 

853.62 

TABLE  No.  6. 
TEN- YEAR  TERM  INSURANCE,  NET  ANNUAL  PREMIUM  $106.03. 


YEAR. 

Net  Reserve 
or  Accumulated 
Deposits  Being 
Self-  Insurance. 

Net  Amount  of 
Insurance  at 
Risk  or  Carried 
by  the  Company. 

Tabular  Cost 
Each  Year 
to  Insure 
$10,000  for 
the  Year. 

Tabular  Cost  to 
Insure  Net 
Amount  at  Risk 
which  is  also 
the  Full  Legal 
and  Mathemat- 
tical  Insurance 
Reserve. 

Deposit 
Portion  of 
Annual 
Premium 
Which  is 
Merely  for 
Accumu- 
lation. 

$12  45 

$9,987.55 

$94.18 

$94.06 

$tl  97 

2 

9,976.63 

96.23 

96.00 

10.03 

32.37 

9,967.63 

9858 

98.26 

7-77 

H 

9,960.82 

IOI.I3 

100.73 

5.30 

43.2O 

9,956.80 

104.12 

103.  57 

2.37 



6  

44.05 

9,955.95 

107.34 

106.87 

—0.84 

40.95 

9,959.05 

III.I7 

110.72 

g 

33.24 

9,906.76 

115.39 

115.01 

3    gg 

19.99 

9,980  oi 

120.28 

120.04 

I4.OI 

10,000.00 

I26.O2 

126  02 

—  iq.qq 

TABLE  No.  7. 

PURE  ENDOWMENT — AGE  40  AT  ISSTTE — $10,000  PAYABLE  ONLY  IN  CASE  OF  BEING  ALIVE 
AT  THE  END  OF  10  YEARS,  OR  AT  AGE  50. 


YEAR. 

Yearly 
Payments. 

Value  (With- 
out Interest)  of 
$t.oo  Pay  able 
Only  in  Case 
of  Surviving  to 
End  of  Year. 

Fund  at 
Beginning  of 
Year. 

Vame  of  Ditto 
Payable 
Only  in  Case 
of  Survivirg. 

Interest  4%. 

Fund  at  End 
of  Year. 

I.....              

$74.7  5Q 

$1  OOQ  8q 

$747  50 

$754  OQ 

$30  2O 

$78  c;  in 

2                        .           

747  5Q 

I  OIO  II 

I  532.78 

I  548  28 

6l  Q3 

I  6lO  21 

74.7  ZQ 

I  OIO  36 

4  

747.  5Q 

1,010.63 

Q.225.IO 

q  250  38 

iqp  Q7 

q  q8q  75 

747  t;o 

I  OIO  Q^ 

4182  61 

167  °.I 

6 

747  C;Q 

i  on  29 

C.  007  C;Q 

,iU-S.  Uj 

5jre  08 

206  20 

c  q6i  28 

7 

747  5Q 

i  on  70 

6  108  87 

6  180  34 

247  21 

^'•J 

6  427  zz 

8              

747  5Q 

I  OI2  15 

7  175  14 

7  262  32 

2QO  4Q 

7  552  8l 

747  5Q 

I,OI2.67 

8  300  40 

8  405  c;6 

336  22 

8  741  78 

jO 

747  ^Q 

I  013  28 

Q  Qjir  OQ 

11 


Insurance  and  investment  therefore  have  no  necessary  connection — either  one 
may  be  obtained  without  the  other. 

(4).  Pure  insurance,  unmixed  with  banking  or  investment,  involves  the  payment 
of  natural  premiums,  which  inevitably  and  inexorably  increase  with  age.  The  only 
way  to  avoid  these  increasing  rates  is  to  pay  largely  in  excess  of  the  requirements  for 
current  death  claims  in  the  earlier  years,  and  thus  provide  a  fund  upon  which  to  draw 
in  the  later  years — that  is  to  say,  by  combining  investment  with  insurance.  The  first 
is  known  as  the  natural  premium  plan,  the  second  as  the  level  premium  plan.  Prop- 
erly administered,  the  one  is  as  safe  and  as  sound  as  the  other,  as  both  depend  upon 
the  application  of  the  same  laws  of  nature  which  govern  the  rates  of  mortality,  or  the 
probability  of  living  and  dying  in  each  successive  year  of  life.  In  fact,  as  before 
stated,  level  premiums  are  simply  the  commuted  equivalents  of  the  increasing  or 
natural  premiums.  In  both  systems,  the  company  must  alike  be  furnished  with  the 
cost  of  insuring  the  net  amount  at  risk  at  the  actual  age  attained  on  each  and  every 
policy  in  force.  This  cost  is  independent  of  the  form  of  policy  contract,  the  age  at 
issue,  or  the  scale  of  premium  charged.  This  cost,  as  previously  stated,  may  be  fur- 
nished either  by  direct,  present  payments,  as  by  natural  premiums,  or  partly  by  direct 
present  payments,  and  partly  by  drawing  upon  the  investment  reserve  or  accumulated 
deposits,  a  fund  contributed  by  the  policyholder  for  this  express  purpose. 

There  are  only  two  sound  systems  of  life  insurance;  the  one  by  natural  premiums, 
increasing  each  year  as  a  man  grows  older ;  the  other,  by  level  premiums,  which 
necessitate  investments  or  accumulated  payments  largely  in  excess  during  the 
earlier  years  to  meet  the  deficiencies  of  the  uniform,  unchanging  premiums  in  later 
years.  The  attempts  by  so  many  co-operative  or  assessment  companies  to  furnish  in- 
surance by  assessments  based  upon  the  age  at  entry,  and  which  rates  do  not  increase 
with  age  must  inevitably  result  in  disappointment  and  disaster,  Natural  laws  may  not 
be  violated  with  impunity. 

SHEPPARD  ROMANS. 

NEW  YORK,  May  10,  1888. 


12 


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